# Gil Schmidt Power Centrality Index

#### Definition

For graph G=(V,E), let R(v,G) be the set of vertices reachable by v in V \ v. Then the Gil-Schmidt power index is defined as

C_GS(v) = sum( 1/d(v,i), i in R(v,G) )/|R(v,G)|,

where d(v,i) is the geodesic distance from v to i in G; the index is taken to be 0 for isolates. The measure takes a value of 1 when v is adjacent to all reachable vertices, and approaches 0 as the distance from v to each vertex approaches infinity. (For finite N=|V|, the minimum value is 0 if v is an isolate, and otherwise 1/(N-1).)

where d(v,i) is the geodesic distance from v to i in G; the index is taken to be 0 for isolates. The measure takes a value of 1 when v is adjacent to all reachable vertices, and approaches 0 as the distance from v to each vertex approaches infinity. (For finite N=|V|, the minimum value is 0 if v is an isolate, and otherwise 1/(N-1).)

#### Requirements

#### Software

#### References

- Gil, J. and Schmidt, S. (1996). “The Origin of the Mexican Network of Power”. Proceedings of the International Social Network Conference, Charleston, SC, 22-25.
- Sinclair, P.A., 2009. Network centralization with the Gil Schmidt power centrality index. Social Networks, 31(3), pp.214-219. DOI: 10.1016/j.socnet.2009.04.004